Disordered ground states for classical discrete-state problems in one dimension

It is known that one-dimensional lattice problems with a discrete, finite set of states per site ''generically'' have periodic ground states (GSs). We consider slightly less generic cases, in which the Hamiltonian is constrained by either spin (S) or spatial (I) inversion symmetry (or both). We show that such constraints give rise to the possibility of disordered GSs over a finite fraction of the coupling-parameter space-that is, without invoking any nongeneric ''fine tuning'' of coupling constants, beyond that arising from symmetry. We find that such disordered GSs can arise for many values of the number of states k at each site and the range r of the interaction. The Ising (k=2) case is the least prone to disorder: I symmetry allows for disordered GSs (without line tuning) only for r greater than or equal to 5, while S symmetry ''never'' gives rise to disordered GSs.